A proposal for using the ensemble approach to understand genetic regulatory networks

Understanding the genetic regulatory network comprising genes, RNA, proteins and the network connections and dynamical control rules among them, is a major task of contemporary systems biology. I focus here on the use of the ensemble approach to find one or more well-defined ensembles of model networks whose statistical features match those of real cells and organisms. Such ensembles should help explain and predict features of real cells and organisms. More precisely, an ensemble of model networks is defined by constraints on the "wiring diagram" of regulatory interactions, and the "rules" governing the dynamical behavior of regulated components of the network. The ensemble consists of all networks consistent with those constraints. Here I discuss ensembles of random Boolean networks, scale free Boolean networks, "medusa" Boolean networks, continuous variable networks, and others. For each ensemble, M statistical features, such as the size distribution of avalanches in gene activity changes unleashed by transiently altering the activity of a single gene, the distribution in distances between gene activities on different cell types, and others, are measured. This creates an M-dimensional space, where each ensemble corresponds to a cluster of points or distributions. Using current and future experimental techniques, such as gene arrays, these M properties are to be measured for real cells and organisms, again yielding a cluster of points or distributions in the M-dimensional space. The procedure then finds ensembles close to those of real cells and organisms, and hill climbs to attempt to match the observed M features. Thus obtains one or more ensembles that should predict and explain many features of the regulatory networks in cells and organisms.     

Discrete logic modelling as a means to link protein signalling networks with functional analysis of mammalian signal transduction

Large‐scale protein signalling networks are useful for exploring complex biochemical pathways but do not reveal how pathways respond to specific stimuli. Such specificity is critical for understanding disease and designing drugs. Here we describe a computational approach—implemented in the free CNO software—for turning signalling networks into logical models and calibrating the models against experimental data. When a literature‐derived network of 82 proteins covering the immediate‐early responses of human cells to seven cytokines was modelled, we found that training against experimental data dramatically increased predictive power, despite the crudeness of Boolean approximations, while significantly reducing the number of interactions. Thus, many interactions in literature‐derived networks do not appear to be functional in the liver cells from which we collected our data. At the same time, CNO identified several new interactions that improved the match of model to data. Although missing from the starting network, these interactions have literature support. Our approach, therefore, represents a means to generate predictive, cell‐type‐specific models of mammalian signalling from generic protein signalling networks.     

Simulated evolution of protein-protein interaction networks with realistic topology

We model the evolution of eukaryotic protein-protein interaction (PPI) networks. In our model, PPI networks evolve by two known biological mechanisms: (1) Gene duplication, which is followed by rapid diversification of duplicate interactions. (2) Neofunctionalization, in which a mutation leads to a new interaction with some other protein. Since many interactions are due to simple surface compatibility, we hypothesize there is an increased likelihood of interacting with other proteins in the target protein's neighborhood. We find good agreement of the model on 10 different network properties compared to high-confidence experimental PPI networks in yeast, fruit flies, and humans. Key findings are: (1) PPI networks evolve modular structures, with no need to invoke particular selection pressures. (2) Proteins in cells have on average about 6 degrees of separation, similar to some social networks, such as human-communication and actor networks. (3) Unlike social networks, which have a shrinking diameter (degree of maximum separation) over time, PPI networks are predicted to grow in diameter. (4) The model indicates that evolutionarily old proteins should have higher connectivities and be more centrally embedded in their networks. This suggests a way in which present-day proteomics data could provide insights into biological evolution.     

Improved coarse‐grained model for studying sequence dependent phase separation of disordered proteins

We present improvements to the hydropathy scale (HPS) coarse‐grained (CG) model for simulating sequence‐specific behavior of intrinsically disordered proteins (IDPs), including their liquid–liquid phase separation (LLPS). The previous model based on an atomistic hydropathy scale by Kapcha and Rossky (KR scale) is not able to capture some well‐known LLPS trends such as reduced phase separation propensity upon mutations (R‐to‐K and Y‐to‐F). Here, we propose to use the Urry hydropathy scale instead, which was derived from the inverse temperature transitions in a model polypeptide with guest residues X. We introduce two free parameters to shift (Δ) and scale (µ) the overall interaction strengths for the new model (HPS‐Urry) and use the experimental radius of gyration for a diverse group of IDPs to find their optimal values. Interestingly, many possible (Δ, µ) combinations can be used for typical IDPs, but the phase behavior of a low‐complexity (LC) sequence FUS is only well described by one of these models, which highlights the need for a careful validation strategy based on multiple proteins. The CG HPS‐Urry model should enable accurate simulations of protein LLPS and provide a microscopically detailed view of molecular interactions.     

Computed free energy differences between point mutations in a collagen-like peptide

We studied the results of mutating alanine --> glycine at three positions of a collagen-like peptide in an effort to develop a computational method for predicting the energetic and structural effects of a single point genetic mutation in collagen, which is associated with the clinical diagnosis of Osteogenesis Imperfecta (OI). The differences in free energy of denaturation were calculated between the collagen-like peptides [(POG)(4)(POA)(POG)(4)](3) and [(POG)(10)](3) (POG: proline-hydroxyproline-glycine).* Our computational results, which suggest significant destabilization of the collagen-like triple-helix upon the glycine --> alanine mutations, correlate very well with the experimental free energies of denaturation. The robustness of our collagen-like peptide model is shown by its reproduction of experimental results with both different simulation paths and different lengths of the model peptide. The individual free energy for each alanine --> glycine mutation (and the reverse free energy, glycine --> alanine mutation) in the collagen-like peptide has been calculated. We find that the first alanine introduced into the triple helix causes a very large destabilization of the helix, but the last alanine introduced into the same position of an adjacent chain causes a very small change in the peptide stability. Thus, our results demonstrate that each mutation does not contribute equally to the free energy. We find that the sum of the calculated individual residues' free energy can accurately model the experimental free energy for the whole peptide.     

Invariant properties in coevolutionary networks of plant–animal interactions

Plant–animal mutualistic networks are interaction webs consisting of two sets of entities, plant and animal species, whose evolutionary dynamics are deeply influenced by the outcomes of the interactions, yielding a diverse array of coevolutionary processes. These networks are two‐mode networks sharing many common properties with others such as food webs, social, and abiotic networks. Here we describe generalized patterns in the topology of 29 plant–pollinator and 24 plant–frugivore networks in natural communities. Scale‐free properties have been described for a number of biological, social, and abiotic networks; in contrast, most of the plant–animal mutualistic networks (65.6%) show species connectivity distributions (number of links per species) with a power‐law regime but decaying as a marked cut‐off, i.e. truncated power‐law or broad‐scale networks and few (22.2%) show scale‐invariance. We hypothesize that plant–animal mutualistic networks follow a build‐up process similar to complex abiotic nets, based on the preferential attachment of species. However, constraints in the addition of links such as morphological mismatching or phenological uncoupling between mutualistic partners, restrict the number of interactions established, causing deviations from scale‐invariance. This reveals generalized topological patterns characteristic of self‐organized complex systems. Relative to scale‐invariant networks, such constraints may confer higher robustness to the loss of keystone species that are the backbone of these webs.     

Patterns of genetic variation in populations of infectious agents

Background  

The analysis of genetic variation in populations of infectious agents may help us understand their epidemiology and evolution. Here we study a model for assessing the levels and patterns of genetic diversity in populations of infectious agents. The population is structured into many small subpopulations, which correspond to their hosts, that are connected according to a specific type of contact network. We considered different types of networks, including fully connected networks and scale free networks, which have been considered as a model that captures some properties of real contact networks. Infectious agents transmit between hosts, through migration, where they grow and mutate until elimination by the host immune system.     

Reverse engineering discrete dynamical systems from data sets with random input vectors.

Recently a new algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is based on methods from computational algebra and finds most parsimonious models for a given data set. We derive mathematically rigorous estimates for the expected amount of data needed by this algorithm to find the correct model. In particular, we demonstrate that for one type of input parameter (graded term orders), the expected data requirements scale polynomially with the number n of chemicals in the network, while for another type of input parameters (randomly chosen lex orders) this number scales exponentially in n. We also show that, for a modification of the algorithm, the expected data requirements scale as the logarithm of n.     

Detecting and Removing Inconsistencies between Experimental Data and Signaling Network Topologies Using Integer Linear Programming on Interaction Graphs

Cross-referencing experimental data with our current knowledge of signaling network topologies is one central goal of mathematical modeling of cellular signal transduction networks. We present a new methodology for data-driven interrogation and training of signaling networks. While most published methods for signaling network inference operate on Bayesian, Boolean, or ODE models, our approach uses integer linear programming (ILP) on interaction graphs to encode constraints on the qualitative behavior of the nodes. These constraints are posed by the network topology and their formulation as ILP allows us to predict the possible qualitative changes (up, down, no effect) of the activation levels of the nodes for a given stimulus. We provide four basic operations to detect and remove inconsistencies between measurements and predicted behavior: (i) find a topology-consistent explanation for responses of signaling nodes measured in a stimulus-response experiment (if none exists, find the closest explanation); (ii) determine a minimal set of nodes that need to be corrected to make an inconsistent scenario consistent; (iii) determine the optimal subgraph of the given network topology which can best reflect measurements from a set of experimental scenarios; (iv) find possibly missing edges that would improve the consistency of the graph with respect to a set of experimental scenarios the most. We demonstrate the applicability of the proposed approach by interrogating a manually curated interaction graph model of EGFR/ErbB signaling against a library of high-throughput phosphoproteomic data measured in primary hepatocytes. Our methods detect interactions that are likely to be inactive in hepatocytes and provide suggestions for new interactions that, if included, would significantly improve the goodness of fit. Our framework is highly flexible and the underlying model requires only easily accessible biological knowledge. All related algorithms were implemented in a freely available toolbox SigNetTrainer making it an appealing approach for various applications.     

Preferential Duplication of Intermodular Hub Genes: An Evolutionary Signature in Eukaryotes Genome Networks

Whole genome protein-protein association networks are not random and their topological properties stem from genome evolution mechanisms. In fact, more connected, but less clustered proteins are related to genes that, in general, present more paralogs as compared to other genes, indicating frequent previous gene duplication episodes. On the other hand, genes related to conserved biological functions present few or no paralogs and yield proteins that are highly connected and clustered. These general network characteristics must have an evolutionary explanation. Considering data from STRING database, we present here experimental evidence that, more than not being scale free, protein degree distributions of organisms present an increased probability for high degree nodes. Furthermore, based on this experimental evidence, we propose a simulation model for genome evolution, where genes in a network are either acquired de novo using a preferential attachment rule, or duplicated with a probability that linearly grows with gene degree and decreases with its clustering coefficient. For the first time a model yields results that simultaneously describe different topological distributions. Also, this model correctly predicts that, to produce protein-protein association networks with number of links and number of nodes in the observed range for Eukaryotes, it is necessary 90% of gene duplication and 10% of de novo gene acquisition. This scenario implies a universal mechanism for genome evolution.     

EXTINCTIONS IN HETEROGENEOUS ENVIRONMENTS AND THE EVOLUTION OF MODULARITY

Extinctions of local subpopulations are common events in nature. Here, we ask whether such extinctions can affect the design of biological networks within organisms over evolutionary timescales. We study the impact of extinction events on modularity of biological systems, a common architectural principle found on multiple scales in biology. As a model system, we use networks that evolve toward goals specified as desired input–output relationships. We use an extinction–recolonization model, in which metapopulations occupy and migrate between different localities. Each locality displays a different environmental condition (goal), but shares the same set of subgoals with other localities. We find that in the absence of extinction events, the evolved computational networks are typically highly optimal for their localities with a nonmodular structure. In contrast, when local populations go extinct from time to time, we find that the evolved networks are modular in structure. Modular circuitry is selected because of its ability to adapt rapidly to the conditions of the free niche following an extinction event. This rapid adaptation is mainly achieved through genetic recombination of modules between immigrants from neighboring local populations. This study suggests, therefore, that extinctions in heterogeneous environments promote the evolution of modular biological network structure, allowing local populations to effectively recombine their modules to recolonize niches.     

Constitutive Model of Erythrocyte Membranes with Distributions of Spectrin Orientations and Lengths

We present an analytical hyperelastic constitutive model of the red blood cell (erythrocyte) membrane based on recently improved characterizations of density and microscopic structure of its spectrin network from proteomics and cryo-electron tomography. The model includes distributions of both orientations and natural lengths of spectrin and updated copy numbers of proteins. By applying finite deformation to the spectrin network, we obtain the total free energy and stresses in terms of invariants of shear and area deformation. We generalize an expression of the initial shear modulus, which is independent of the number of molecular orientations within the network and also derive a simplified version of the model. We apply the model and its simplified version to analyze micropipette aspiration computationally and analytically and explore the effect of local cytoskeletal density change. We also explore the discrepancies among shear modulus values measured using different experimental techniques reported in the literature. We find that the model exhibits hardening behavior and can explain many of these discrepancies. Moreover, we find that the distribution of natural lengths plays a crucial role in the hardening behavior when the correct copy numbers of proteins are used. The initial shear modulus values we obtain using our current model (5.9–15.6 pN/μm) are close to the early estimates (6–9 pN/μm). This new, to our knowledge, constitutive model establishes a direct connection between the molecular structure of spectrin networks and constitutive laws and also defines a new picture of a much denser spectrin network than assumed in prior studies.     

When shared concept cells support associations: Theory of overlapping memory engrams

Assemblies of neurons, called concepts cells, encode acquired concepts in human Medial Temporal Lobe. Those concept cells that are shared between two assemblies have been hypothesized to encode associations between concepts. Here we test this hypothesis in a computational model of attractor neural networks. We find that for concepts encoded in sparse neural assemblies there is a minimal fraction c_min of neurons shared between assemblies below which associations cannot be reliably implemented; and a maximal fraction c_max of shared neurons above which single concepts can no longer be retrieved. In the presence of a periodically modulated background signal, such as hippocampal oscillations, recall takes the form of association chains reminiscent of those postulated by theories of free recall of words. Predictions of an iterative overlap-generating model match experimental data on the number of concepts to which a neuron responds.     

The dimensionality of ecological networks

How many dimensions (trait‐axes) are required to predict whether two species interact? This unanswered question originated with the idea of ecological niches, and yet bears relevance today for understanding what determines network structure. Here, we analyse a set of 200 ecological networks, including food webs, antagonistic and mutualistic networks, and find that the number of dimensions needed to completely explain all interactions is small ( < 10), with model selection favouring less than five. Using 18 high‐quality webs including several species traits, we identify which traits contribute the most to explaining network structure. We show that accounting for a few traits dramatically improves our understanding of the structure of ecological networks. Matching traits for resources and consumers, for example, fruit size and bill gape, are the most successful combinations. These results link ecologically important species attributes to large‐scale community structure.     

Functional Inference of Complex Anatomical Tendinous Networks at a Macroscopic Scale via Sparse Experimentation

In systems and computational biology, much effort is devoted to functional identification of systems and networks at the molecular-or cellular scale. However, similarly important networks exist at anatomical scales such as the tendon network of human fingers: the complex array of collagen fibers that transmits and distributes muscle forces to finger joints. This network is critical to the versatility of the human hand, and its function has been debated since at least the 16^th century. Here, we experimentally infer the structure (both topology and parameter values) of this network through sparse interrogation with force inputs. A population of models representing this structure co-evolves in simulation with a population of informative future force inputs via the predator-prey estimation-exploration algorithm. Model fitness depends on their ability to explain experimental data, while the fitness of future force inputs depends on causing maximal functional discrepancy among current models. We validate our approach by inferring two known synthetic Latex networks, and one anatomical tendon network harvested from a cadaver''s middle finger. We find that functionally similar but structurally diverse models can exist within a narrow range of the training set and cross-validation errors. For the Latex networks, models with low training set error [<4%] and resembling the known network have the smallest cross-validation errors [∼5%]. The low training set [<4%] and cross validation [<7.2%] errors for models for the cadaveric specimen demonstrate what, to our knowledge, is the first experimental inference of the functional structure of complex anatomical networks. This work expands current bioinformatics inference approaches by demonstrating that sparse, yet informative interrogation of biological specimens holds significant computational advantages in accurate and efficient inference over random testing, or assuming model topology and only inferring parameters values. These findings also hold clues to both our evolutionary history and the development of versatile machines.     

A Flexible Mathematical Model Platform for Studying Branching Networks: Experimentally Validated Using the Model Actinomycete,Streptomyces coelicolor

Branching networks are ubiquitous in nature and their growth often responds to environmental cues dynamically. Using the antibiotic-producing soil bacterium Streptomyces as a model we have developed a flexible mathematical model platform for the study of branched biological networks. Streptomyces form large aggregates in liquid culture that can impair industrial antibiotic fermentations. Understanding the features of these could aid improvement of such processes. The model requires relatively few experimental values for parameterisation, yet delivers realistic simulations of Streptomyces pellet and is able to predict features, such as the density of hyphae, the number of growing tips and the location of antibiotic production within a pellet in response to pellet size and external nutrient supply. The model is scalable and will find utility in a range of branched biological networks such as angiogenesis, plant root growth and fungal hyphal networks.     

Stability of Metabolic Correlations under Changing Environmental Conditions in Escherichia coli – A Systems Approach

Background

Biological systems adapt to changing environments by reorganizing their cellular and physiological program with metabolites representing one important response level. Different stresses lead to both conserved and specific responses on the metabolite level which should be reflected in the underlying metabolic network.

Methodology/Principal Findings

Starting from experimental data obtained by a GC-MS based high-throughput metabolic profiling technology we here develop an approach that: (1) extracts network representations from metabolic condition-dependent data by using pairwise correlations, (2) determines the sets of stable and condition-dependent correlations based on a combination of statistical significance and homogeneity tests, and (3) can identify metabolites related to the stress response, which goes beyond simple observations about the changes of metabolic concentrations. The approach was tested with Escherichia coli as a model organism observed under four different environmental stress conditions (cold stress, heat stress, oxidative stress, lactose diauxie) and control unperturbed conditions. By constructing the stable network component, which displays a scale free topology and small-world characteristics, we demonstrated that: (1) metabolite hubs in this reconstructed correlation networks are significantly enriched for those contained in biochemical networks such as EcoCyc, (2) particular components of the stable network are enriched for functionally related biochemical pathways, and (3) independently of the response scale, based on their importance in the reorganization of the correlation network a set of metabolites can be identified which represent hypothetical candidates for adjusting to a stress-specific response.

Conclusions/Significance

Network-based tools allowed the identification of stress-dependent and general metabolic correlation networks. This correlation-network-based approach does not rely on major changes in concentration to identify metabolites important for stress adaptation, but rather on the changes in network properties with respect to metabolites. This should represent a useful complementary technique in addition to more classical approaches.     

Comparison of experimental and theoretical data on hydrogen-deuterium exchange for ten globular proteins

The number of protons available for hydrogen-deuterium exchange was predicted for ten globular proteins using a method described elsewhere by the authors. The average number of protons replaced by deuterium was also determined by mass spectrometry of the intact proteins in their native conformations. Based on these data, we find that two models proposed earlier agree with each other in estimation of the number of protons replaced by deuterium. Using a model with a probability scale for hydrogen bond formation, we estimated a number of protons replaced by deuterium that is close to the experimental data for long-term incubation in D₂O (24 h). Using a model based on estimations with a scale of the expected number of contacts in globular proteins there is better agreement with the experimental data obtained for a short period of incubation in D₂O (15 min). Therefore, the former model determines weakly fluctuating parts of a protein that are in contact with solvent only for a small fraction of the time. The latter model (based on the scale of expected number of contacts) predicts either flexible parts of a protein chain exposed to interactions with solvent or disordered parts of the protein.     

GraphCrunch 2: Software tool for network modeling,alignment and clustering

Background  

Recent advancements in experimental biotechnology have produced large amounts of protein-protein interaction (PPI) data. The topology of PPI networks is believed to have a strong link to their function. Hence, the abundance of PPI data for many organisms stimulates the development of computational techniques for the modeling, comparison, alignment, and clustering of networks. In addition, finding representative models for PPI networks will improve our understanding of the cell just as a model of gravity has helped us understand planetary motion. To decide if a model is representative, we need quantitative comparisons of model networks to real ones. However, exact network comparison is computationally intractable and therefore several heuristics have been used instead. Some of these heuristics are easily computable "network properties," such as the degree distribution, or the clustering coefficient. An important special case of network comparison is the network alignment problem. Analogous to sequence alignment, this problem asks to find the "best" mapping between regions in two networks. It is expected that network alignment might have as strong an impact on our understanding of biology as sequence alignment has had. Topology-based clustering of nodes in PPI networks is another example of an important network analysis problem that can uncover relationships between interaction patterns and phenotype.